Problem: Simplify the following expression: $\dfrac{81y^2}{54y^5}$ You can assume $y \neq 0$.
Answer: $ \dfrac{81y^2}{54y^5} = \dfrac{81}{54} \cdot \dfrac{y^2}{y^5} $ To simplify $\frac{81}{54}$ , find the greatest common factor (GCD) of $81$ and $54$ $81 = 3 \cdot 3 \cdot 3 \cdot 3$ $54 = 2 \cdot 3 \cdot 3 \cdot 3$ $ \mbox{GCD}(81, 54) = 3 \cdot 3 \cdot 3 = 27 $ $ \dfrac{81}{54} \cdot \dfrac{y^2}{y^5} = \dfrac{27 \cdot 3}{27 \cdot 2} \cdot \dfrac{y^2}{y^5} $ $\phantom{ \dfrac{81}{54} \cdot \dfrac{2}{5}} = \dfrac{3}{2} \cdot \dfrac{y^2}{y^5} $ $ \dfrac{y^2}{y^5} = \dfrac{y \cdot y}{y \cdot y \cdot y \cdot y \cdot y} = \dfrac{1}{y^3} $ $ \dfrac{3}{2} \cdot \dfrac{1}{y^3} = \dfrac{3}{2y^3} $